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arxiv: 2605.28319 · v1 · pith:DH45XSVEnew · submitted 2026-05-27 · 🧮 math-ph · cond-mat.stat-mech· math.MP· math.PR

Dissipative Spectral Form Factor of the Complex Elliptic Ginibre Ensemble across Various Non-Hermiticity Regimes

Gernot Akemann , Sung-Soo Byun , Seungjoon Oh This is my paper

Pith reviewed 2026-06-29 09:54 UTC · model grok-4.3

classification 🧮 math-ph cond-mat.stat-mechmath.MPmath.PR
keywords dissipative spectral form factorelliptic Ginibre ensemblenon-Hermiticityrandom matricesspectral statisticsdip-ramp-plateauasymptotic analysis
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The pith

The dissipative spectral form factor of the elliptic Ginibre ensemble shows a dip-ramp-plateau whose ramp shape depends on the joint scaling of time and non-Hermiticity in the large-N limit.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies the dissipative spectral form factor evaluated at complex times for the complex elliptic Ginibre ensemble, whose matrices are controlled by a non-Hermiticity parameter ranging from fully non-Hermitian to nearly Hermitian. It examines the limit of infinite matrix size while letting the time scale as a power of N and the distance to Hermiticity scale as another power of N. This produces explicit leading asymptotics for both the disconnected and connected pieces of the form factor in every combination of those exponents. The results map the full dip-ramp-plateau shape, locate the dip time and Heisenberg time, and isolate an intermediate scaling window in which the form factor continuously crosses over from non-Hermitian to Hermitian behavior.

Core claim

As N tends to infinity under the scalings T = O(N^γ) and 1-τ = O(N^{-α}) for all γ ≥ 0 and α ≥ 0, both the disconnected and connected components of the DSFF admit precise asymptotic expansions that fully characterize the dip-ramp-plateau structure together with its dip time and Heisenberg time; the mesoscopic window α ∈ (0,1) produces an interpolation between the DSFF of non-Hermitian matrices and the ordinary spectral form factor of Hermitian ensembles, while the ramp itself takes quadratic, linear, or intermediate form according to the values of the scaling exponents.

What carries the argument

The dissipative spectral form factor (DSFF) at complex time Te^{iθ} for the elliptic Ginibre ensemble with non-Hermiticity parameter τ, analyzed under joint power-law scalings of time and 1-τ.

If this is right

  • The dip time and Heisenberg time receive explicit characterizations in every regime.
  • The ramp of the DSFF takes quadratic, linear, or intermediate shape depending on the pair of scaling exponents.
  • The mesoscopic regime α ∈ (0,1) produces a continuous interpolation between fully non-Hermitian and Hermitian spectral statistics.
  • Both the disconnected and connected parts of the DSFF receive complete asymptotic descriptions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same scaling approach may uncover analogous crossover regimes in other non-Hermitian ensembles.
  • Open quantum systems with tunable dissipation strength could exhibit the predicted change in ramp shape when their spectral statistics are measured.
  • The phase diagram supplies concrete transition lines that finite-N simulations can test directly.

Load-bearing premise

The chosen power-law scalings for time and the deviation from Hermiticity are the ones that exhaust all relevant asymptotic regimes of the DSFF.

What would settle it

A numerical computation of the DSFF for the elliptic Ginibre ensemble at large but finite N, performed inside one of the scaling regimes, that deviates from the predicted leading asymptotic expression.

Figures

Figures reproduced from arXiv: 2605.28319 by Gernot Akemann, Seungjoon Oh, Sung-Soo Byun.

Figure 1
Figure 1. Figure 1: Phase diagrams of the DSFF in the (α, γ)–plane, where the non-Hermiticity and time scalings are given by 1 − τ = N −α and T = NγT, with T > 0 fixed. The Thouless time TTh and the generalised Heisenberg time TH are indicated on the figure. (A) Dip–ramp– plateau structure of the DSFF across different time scales. (B) Universality crossover of the DSFF, showing GUE-type behaviour for γ ≤ α and GinUE-type beha… view at source ↗
Figure 2
Figure 2. Figure 2: DSFF in the regime of strong non-Hermiticity (α = 0) with τ = 0.3 and θ = π/6. Here, all axes are on a logarithmic scale. Top: DSFF for N = 64 (green, dashed), together with the analytic limits for γ = 0 (red, solid) and γ = 1 2 (blue, solid). Bottom left: DSFF for γ = 0 for varying N, compared with the analytic limit. Bottom right: DSFF for γ = 1 2 for varying N, compared with the analytic limit. We also … view at source ↗
Figure 3
Figure 3. Figure 3: The plots illustrate the DSFF in the mesoscopic non-Hermiticity regime with pa￾rameters N = 214 , θ = π/6, and γ = 0.6, in all 3 figures. Accordingly, in case (A) with α < γ, one observes the quadratic ramp κT 2 2 , whereas in case (C) with α > γ, the behaviour is given by the linear ramp 2t π . In the critical case (B), where α = γ, the resulting behaviour is a combination of these two regimes; see (2.15)… view at source ↗
Figure 4
Figure 4. Figure 4: Decomposition of the positive real axis into the four asymptotic regimes for the Laguerre polynomials. Recall that fN (x) is defined in (3.16). We first derive its uniform asymptotic behaviour in all relevant regimes, with sufficient precision for the subsequent analysis. In particular, these estimates will allow us to obtain the asymptotic behaviours of ΨN and ρN via Proposition 3.4. For this purpose, we … view at source ↗
Figure 5
Figure 5. Figure 5: Each plot compares the asymptotic behaviour of fN in the corresponding regime. In the regimes I, II, and III, the dotted, dot-dashed, and dashed curves represent the remainder obtained after subtracting the expansion up to the second term, for N = 8, 16, and 32 respec￾tively, while the black solid curve shows the third-order term. In the regime II, the oscillatory component of the third-order term is furth… view at source ↗
read the original abstract

We study the dissipative spectral form factor (DSFF) at complex time $T e^{i\theta}$ for the complex elliptic Ginibre ensemble with non-Hermiticity parameter $\tau \in [0,1)$. As the matrix dimension $N \to \infty$, we consider the natural scalings in both the time variable and the non-Hermiticity parameter, namely $T = O(N^\gamma)$ and $1 - \tau = O(N^{-\alpha})$. For all regimes $\gamma \ge 0$ and $\alpha \ge 0$, we derive the precise asymptotic behaviour of both the disconnected and connected components of the DSFF. In particular, we explicitly characterise the dip--ramp--plateau structure, including the dip time and the Heisenberg time. In addition, we identify the mesoscopic regime $\alpha \in (0,1)$, which interpolates between the behaviour of the DSFF of non-Hermitian random matrices and the spectral form factor (SFF) of Hermitian ensembles. We further provide an explicit description of the phase diagram, in which the ramp exhibits quadratic, linear, or intermediate behaviour depending on the scaling parameters.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript derives the precise large-N asymptotic behavior of both the disconnected and connected components of the dissipative spectral form factor (DSFF) for the complex elliptic Ginibre ensemble at complex time T e^{iθ}. It considers the scalings T = O(N^γ) and 1−τ = O(N^{-α}) for all γ ≥ 0 and α ≥ 0, explicitly characterizing the dip–ramp–plateau structure (including dip time and Heisenberg time), identifying the mesoscopic regime α ∈ (0,1) that interpolates between non-Hermitian DSFF and Hermitian SFF, and providing a phase diagram in which the ramp is quadratic, linear, or intermediate depending on the parameters.

Significance. If the derivations are correct, the work supplies a complete phase diagram for DSFF across non-Hermiticity regimes, including an explicit interpolation to the Hermitian limit. This is a substantive contribution to random-matrix theory for open quantum systems, particularly for understanding spectral correlations at complex times.

major comments (1)
  1. [Abstract] Abstract: the claim that the scalings T = O(N^γ) and 1−τ = O(N^{-α}) exhaust all relevant asymptotic regimes for the DSFF (both disconnected and connected) rests on the unstated assumption that the argument θ of the complex time does not introduce an independent N-dependent scale. No justification is given that crossovers arising from Im(log T) ~ N^β are either absent or already covered by the existing (γ,α) plane; if such scales exist, the phase diagram and the interpolation for α ∈ (0,1) would be incomplete.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comment regarding the abstract. We address the point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that the scalings T = O(N^γ) and 1−τ = O(N^{-α}) exhaust all relevant asymptotic regimes for the DSFF (both disconnected and connected) rests on the unstated assumption that the argument θ of the complex time does not introduce an independent N-dependent scale. No justification is given that crossovers arising from Im(log T) ~ N^β are either absent or already covered by the existing (γ,α) plane; if such scales exist, the phase diagram and the interpolation for α ∈ (0,1) would be incomplete.

    Authors: We agree that an N-dependent scaling of the phase θ could in principle define additional regimes. In the present work θ is held fixed (O(1)), independent of N; this is the conventional choice when studying DSFF at complex times T e^{iθ}. With θ fixed, the leading large-N asymptotics of both the disconnected and connected DSFF are controlled by the magnitude T ~ N^γ and the non-Hermiticity parameter 1−τ ~ N^{-α}, and the phase factors arising from θ do not generate new crossover scales. If θ were allowed to scale as N^β, a separate scaling analysis would indeed be required, but that lies outside the scope of the regimes we consider. We will add an explicit statement in the abstract and introduction clarifying that θ is fixed and independent of N, thereby justifying completeness of the (γ,α) phase diagram for the fixed-θ case. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses standard large-N analysis on independent ensemble properties

full rationale

The abstract states that the authors derive the DSFF asymptotics for the given scalings of T and τ directly from the complex elliptic Ginibre ensemble. No quoted steps reduce the claimed results to fitted parameters renamed as predictions, self-definitional loops, or load-bearing self-citations whose content is itself unverified. The phase diagram and dip-ramp-plateau characterization are presented as outputs of the large-N limit applied to the ensemble's spectral statistics, which are external to the DSFF expressions themselves. The choice of scalings is labeled 'natural' but does not enter the derivation as a fitted input that forces the output by construction. This is the expected non-circular case for a mathematical physics derivation grounded in explicit asymptotic analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Relies on standard large-N limits and random-matrix asymptotics for Ginibre ensembles; no free parameters, new entities, or ad-hoc axioms stated in the abstract.

axioms (1)
  • domain assumption The large-N limit with the stated scalings T = O(N^γ) and 1-τ = O(N^{-α}) exists and yields well-defined asymptotics for the DSFF.
    Invoked when taking N → ∞ in the abstract description of regimes.

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